Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 45

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If the arithmetic mean of two positive numbers $a & b (a > b)$ is twice their geometric mean, then $a : b$ is

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Important Questions on Sequences and Series

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 46

If

\sum_{i = 1}^{n} r (r + 1) (2 r + 3) = a n^{4} + b n^{3} + c n^{2} + d n + e,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 47

The value of

\sum_{r = 1}^{n} \frac{1}{\sqrt{a + r x} + \sqrt{a + (r - 1) x}}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 48

The roots of the equation

x^{4} - 8 x^{3} + a x^{2} - b x + 16 = 0,

are positive, if

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 1

If

x, y, z

are in A.P. and

\tan {}^{- 1}x, \tan^{- 1} y

and

\tan^{- 1} z

are also in A.P., then

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 2

If

a_{n} = \frac{3}{4} - {(\frac{3}{4})}^{2} + {(\frac{3}{4})}^{3} + \dots \dots {(- 1)}^{n - 1} {(\frac{3}{4})}^{n}

and

b_{n} = 1 - a_{n}

, then find the minimum natural number

n_{0}

such that

b_{n} > a_{n} \forall n > n_{0}

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 5

Suppose four distinct positive numbers $a_{1}, a_{2}, a_{3}, a_{4}$ are in G.P. Let $b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, b_{3} = b_{2} + a_{3}$ and $b_{4} = b_{3} + a_{4}$ .

STATEMENT - $1$ : The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ are neither in A.P. nor in G.P.

STATEMENT - $2$ : The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ are in H.P.

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 6

If the sum of first

n

terms of an A.P. is

c n^{2}

, then the sum of squares of these

n

terms is

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Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-3>Q 7

Let

S_{k}, k = 1, 2 . . . ., 100,

denote the sum of the infinite geometric series whose first term is

\frac{k - 1}{k!}

and the common ratio is

\frac{1}{k},

then the value of

\frac{100^{2}}{100!} + \sum_{k = 1}^{100} |(k^{2} - 3 k + 1) S_{k}|

is